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C OMPOSITIO M ATHEMATICA

R AYMOND Y. T. W ONG

Homotopy negligible subsets of bundles

Compositio Mathematica, tome 24, n

o

1 (1972), p. 119-128

<http://www.numdam.org/item?id=CM_1972__24_1_119_0>

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(2)

119

HOMOTOPY NEGLIGIBLE SUBSETS OF BUNDLES by

Raymond

Y. T.

Wong

COMPOSITIO MATHEMATICA, Vol. 24, Fasc. 1, 1972, pag. 119-128 Wolters-Noordhoff Publishing

Printed in the Netherlands

1. Introduction

Recently

Eells and

Kuiper [2] have

shown that for any

ANR, locally homotopy negligible

closed subsets are

homotopy negligible.

It is conceiv- able that such results will be

applicable

to

manifolds, particularly

of

infinite-dimension. The purpose of this paper is to

derive,

in the

setting

of

locally

trivial bundle ,a condition that a closed subset of the total space will be

locally homotopy negligible.

A

typical

result is

(for

a more

precise

statement, see Theorem

3)

that for a

locally

trivial bundle

03BE :

X ~ B with base

space B

and fibre F

being

manifolds modeled

on a closed convex subset of some metric

locally

convex

topological

vector space

(MLCTVS),

a closed subset K of the total space X is

locally homotopy negligible

if the restriction of K to each fibre

03BE-1(b)

is

locally homotopy negligible in 03BE-1(b).

All

topological

vector spaces

(TVS)

considered are metric

locally

convex

topological

vector spaces

(LCTVS).

We say that a subset A of a

topological

space X is :

(weakly) homotopy negligible

if the inclusion X-A ~ X is a

(weak) homotopy equivalence;

locally homotopy negligible

if each

point

of A has a fundamental

system

of neighborhoods {U}

such that the inclusion U- A - U is a

homotopy equivalence; strongly homotopy negligible

if for any

neighborhood

U of

X,

the inclusion U- A - U is a

homotopy equivalence;

a Z-set

(that is,

a set with

property Z,

see Anderson

[1 ])

if for each

homotopically

trivial

non-empty

open subset U

of X,

U- A is

non-empty

and

homotopically

trivial.

We say X is a manifold modeled on a space C if X has an open cover-

ing by

sets

homeomorphic

to open subsets of C. For our purpose, a

locally

trivial bundle has at least the

following

structure:

(1)

a

map 03BE : X ~ B,

where X is called the total space, B the base space

and 03BE : X ~ B

the

projection,

(2)

a space F called the

fibre,

(3)

for each b E

B,

there is an open set U of b and a

homeomorphism

ç

of 03BE-1(U)

onto U x F such that the

following diagram

commutes

(3)

03BE-1(b)

is called the fibre over b.

A map

f :

X x Y - X x Z is

X-preserving

if

f(x, y) = (x, z)

for each

(x,y)EXxY.

Absolute retract

(AR),

absolute

neighborhood

retract

(ANR)

are

metric spaces in the sense of Palais

[3].

PROPOSITION 1. Let X be a

Hausdorff

paracompact

manifold

modeled

on a closed convex subset

of

a metrizable

locally

convex

topological

vector

space

(LCTVS)

E. For a closed subset A

of X,

we have the

following implication.

(1)

A is

strongly homotopy negligible

(2)

A is

locally homotopy negligible

~

(5)

A is a Z-set

(3)

A is

homotopy negligible

(4)

A is

weakly homotopy negligible.

PROOF.

(1)

~

(2), (1)

~

(5), (5)

~

(2), (3)

~

(4), (1)

~

(3)

are

trivial. It follows from the

hypothesis

that X is a metrizable space which is

locally

an

ANR,

hence

by [3-Thm. 5],

X is an ANR. For an

ANR,

(4)

~

(3)

is a well-known theorem of J. H. C. Whitehead and

(2)

~

(1)

is the content of the main lemma of Eells and

Kuiper [2].

For

application

of

homotopy negligible

subsets to various manifolds

and the relation of it with

negligible subsets,

we refer to

[3]

and

[4].

2. Statement of the Theorems

Throughout

this

section, by

a

manifold

we shall mean a

paracompact

Hausdorff space modeled on

C,

where C is a closed convex subset of a metric

locally

convex

topological

vector space

(LCTVS)

E. Let R denote

the reals. The

following examples

of C are of interest.

(1)

C = a closed convex subset of E

(2) C = E

(3)

C = a

compact

convex subset of E

(4) C = [-1, 1]n for n = 1, ···, ~

(5)

C =

(- 1, 1)n

for n =

1,...,

oo.

(4)

121

We remark that C is a metrizable AR and X is a metrizable ANR.

The

proofs

of the

following

theorems

(except

for Theorem

3)

will be

given

in section 4.

THEOREM 1. Let X be a

manifold

and R’ a closed interval in R. Let K be

a closed subset of R’ X such that K ~ t X is locally homotopy negligible

in each t x

X,

then K is

strongly homotopy negligible

in R’ x X.

Furthermore,

the same is true when

’locall y homotopy negligible’

is

replaced by

’a Z-set’.

COROLLARY 1. Let X be a Fréchet

manifold

and K a closed set in

(0, 1)

x X such that K n t x X is a Z-set in each t x X. Then K is a Z-set in

(0, 1)

x X.

(See

Theorem 4 for a

stronger result.)

Thus we settle a

question

raised in

[5-8].

THEOREM 2. Let

Xl , X2

be

manifolds

and K a closed subset

of Xl

x

X2

such that K n x x

X2

is

locally homotopy negligible

in x x

X2 for

each

x E

Xl.

Then K is

strongly homotopy negligible

in

Xl

x

X2.

COROLLARY 2. Let X be a

manifold

and K a closed subset

of

s x

X,

where s =

( -1, 1)~,

such that each x x X n K is a Z-set in x x X. Then

K is a Z-set in s x X.

The

following

theorem is a consequence of Theorem 2.

THEOREM 3.

Let 03BE :

X ~ B be a

locally-trivial

bundle with

fibre

F such

that X is paracompact

Hausdorff

and both F and B are

manifolds.

Then a

closed subset K

of

the total space X is

strongly homotopy negligible

in X

if

K

~ 03BE-1 (b)

is

locally homotopy negligible

in each

fibre 03BE-1(b)

over b.

PROOF. It suffices to observe that X is

locally

a

product

of manifolds

Ml

x

M2,

where

Ml, M2

are open subsets of

B,

F

respectively,

such that

K n

(x

x

M2)

is

locally homotopy negligible

in x x

M2

for each x E

Ml.

By

Theorem

2,

K n

Ml

x

M2

is

strongly homotopy negligible

in

Ml

x

M2.

Then we observe that X is an ANR

[3-Thm. 5] and {M1 M2}

form a

neighborhood system

of X. Now

apply proposition

1.

THEOREM 4. Let In =

[-1, 1]n

be the

n-cube,

n =

1, 2, ···, ~,

and let Sn =

(-1, 1)n

be the interior

(or pseudo-interior

in case

of

n =

~) of

I". Let X be a

manifold.

Then a closed subset K

of

In X X is

strongly homotopy negligible

in In x X

if

K n

(x

x

X)

is

locally homotopy negligible

in x x

X for

each x E sn.

Furthermore,

the same is true when we

replace ’locally homotopy negli- gible’ by

’a Z-set’.

THEOREM 5. Let

Cl , C2

be spaces such that

Cl

is a closed convex subset

(5)

of

a metric LCTVS

Ei. Suppose

K is a closed subset

of Cl

x

C2

such that

K n

(x

x

C2 )

is

weakly homotopy negligible

in x x

C2 for

each x E

Cl ,

then K is

homotopy negligible

in

Cl

x

C2.

COROLLARY 3. Let X be a Fréchet space or the Hilbert

cube,

then K is

homotopy negligible

in R x X

if

K n t x X is

ewakly homotopy negligible

in each t x X.

THEOREM 6. Let In =

[0, 1]n

and sn =

(0, 1)n,

n =

1, 2, ···, ~.

Let C

be any closed convex subset

of

a TVS E.

Suppose

K is a closed subset

of

In x C such

that for

each x E

sn,

K n

(x

x

C)

is

weakly homotopy negligible

in x x C. Then K is

homotopy negligible

in In X C.

COROLLARY 4. Let X = a Fréchet space or the Hilbert cube. Then a

closed set K in

[0,

1

] x

X is

homotopy negligible

in

[0, 1]

x X

if K

n

(t

x

X)

is

weakly homotopy negligible

in each t x

X for

0 t 1.

Added in

Proof.

Since this paper was

written,

the author in

[6-Cor 5]

has

generalized

Theorem 3 to include all ANRS.

Precisely,

it states that

Theorem 3 is true when we

replace

X

by

any metric

ANR,

F

by

any metric space which is

locally

an AR

(or manifold)

and B

by

any Hausdorff space. This result is based on a

Lifting

Theorem

[6-Thm 8].

3. Lemmas

LEMMA 1. Let

Ml, M2

be metric spaces and Y an absolute

neighborhood

retract

(ANR).

Let K be a closed subset

of Ml

x

M2 . Suppose f :

K ~

Ml

x Y is a

M1-preserving

map,

then f

can be extended to a

Ml preserving

map

of a neighborhood

U

of K

into

Mi

x Y.

PROOF. Let P :

Ml

x Y ~

Y be the projection. Then Pf :

K ~ Y extends

to a map g of a

neighborhood

U of K into Y. Define F : U ~

Ml

x Y

by F(xl , x2) = (x1 , g(Xl’ X2)).

F is a desired extension.

Let E n denote the Euclidean n-space

R1 R2

X - - - x

Rn

where

Ri

is

the real R. Let

Dn, Sn-1

denote

respectively

the n-ball and

(n-1)-sphere

of E n. We

regard

En as the subset En 0 in

En+1.

Let a b c and consider

[a, c ] as

a closed interval in

Rn

+ 2.

LEMMA 2. Let X be an ANR and K a closed subset

of Rn+2

x X. For

n ~

0,

and i =

1, 2,

suppose

where

I1 = [b, c], I2 = [a, b],

are

Rn+2-preserving

maps such that

(1 ) f1|b Sn-1 = f2|b Sn-1

and

(2) f1|b Dn is homotopic to f2|b Dn in

b X - K modulo

b Sn-1, that is, the

entire

homotopy

agrees

with f1

(6)

123

on

b Sn-1.

Then there is an

Rn+2-preserving

map F

of [a, c ] x Dn

into

Rn+2 X-K such

that

F|I1 Sn-1 ~ c Dn = f1

and

F|I2 Sn-1 ~ a Dn = f2.

PROOF. We note that for n =

0, Sn-1 = 0.

Thus

requirements

on

Sn-1

become irrelevant and

meaningless.

With that modification in mind the

following proof

goes

through

even for this case.

We may assume a

= -1, b = 0

and c = 1. Recall that

I1 = [0, 1 ], 12 = [ -1, 0 ]

and we

regard [ -1, 1 ]

as a subset of

Rn+2.

Define for

i =

1, 2, Rn

+

2-preserving

maps

by

and

It is clear that

Next define a map

ho

of 0 x Sn into 0 x X - K

by

By (1)

of the

hypothesis fl , f2

agree on 0 x

sn-le

Hence

ho

is well-defined.

By

condition

(2)

of the

hypothesis, ho

can be extended to a map h of

Dn+1

into 0 x X-K. Let

Define a

Rn + 2-preserving

map

Since

Rn+2 X-K

is an

ANR, by

Lemma

1,

0 can be extended to a

Rn+2-preserving map 0i

of a

neighborhood

U of A in

En+2

into

Rn+2 X-K.

It is

elementary

to know that there is a

Rn

+

2-preserving map H

of

En + 2

onto itself such that

(1)

H is the

identity

outside of

U, (2)

H is one-to-one

(7)

on

En+2-Dn+1, (3)

H

collapses Dn+1

onto Dn

by H(x1, ···, xn+1)

=

(x1, ···, xn, 0) and (4) H|[-1,1] Sn-1 ~ {-1,1} Dn = identity.

We now

define

a

Rn

+

2-preserving

map F :

[-1, 1] Dn ~ Rn+2 X-K by

F is the desired function.

LEMMA 3. Let E be a metric TVS and u an

n-simplex

in R x E. Let Y be

a metric absolute retract

(AR)

and K a closed subset

of

R x Y such that

K n t x Y is

weakly homotopy negligible

in t x Y

for

each t. Then any

R-preserving

map

of Bd( (J)

into R x Y- K can be extended to an

R-preserv-

ing map of

03C3 into R x Y-K.

PROOF. It is known that a metric AR is contractible. It follows that t x Y- K is

homotopically

trivial for each t. Let à

= boundary

(J,

03C3(t) =

J n t x E and

6(t)

=

boundary 03C3(t).

Let P : R x E - R be the

projection

and let

[a, b]

=

P(03C3), a ~

b. If a =

b,

the lemma follows

immediately

from the

hypothesis.

So suppose a b.

By convexity

each

6(t)

is a

(n -1 )-simplex

for a t b.

Let g : à

~ R x Y-K be a

R-preserving

map. At this

point

we divide

the

argument

into two cases.

CASE 1. n &#x3E; 1.

By hypothesis

of

K,

we have for

each t,

a map gt :

u(t)

~ t x Y-K such that

gt|03C3(t)

= g. For fixed t let A = 03C3 u

6(t)

and define a

R-preserving

map

Gt :

A - t x

Y- K by Gt(x)

=

g(x)

when

x ~ 03C3 and

Gt(x)

=

gt(x)

when x ~

6(t).

Since R x Y-K is an

ANR, by

Lemma

1, Gt

can be extended to a

R-preserving map Gt of

a

neighbor-

hood

Ut

of A into R x Y-K. Since

6(t)

is

compact,

there is an open

interval

(at, bt) containing

t such that

G’(u(s»

n

K = Ø

for all

at ~ s ~ bt.

Since

[a, b]

is

compact,

finite number of

(at, bt)

covers

[a, b].

It follows that there are a finite number of reals a = ao a,

··· am+1 = b and a collection of

R-preserving

maps

{fi}mi=0

such

that for all

i,

where Ai = U{03C3(t)|ai ~ t ~ ai+1}, i = 0, 1, ···, m, and

Since

g(6(a) u 03C3(b))

~ K =

0,

we may assume

f0|03C3(a1) = f1|03C3(a1)

and

fm-1|03C3(am) = fmlu(am).

Consider fl and f2 .

It is clear that

A1, A2

can be

regarded respectively

as

[a1, a2] Dn-1, [a2, a3] Dn-1.

At the level

a2xDn-l

we have two

(8)

125

maps f1, f2 : a2 Dn-1 ~

a2

Y-K such that f1|a2 Sn-2 = f2|a2 Sn-2

= g.

Since

a2 Y-K

is

homotopically trivial,

it follows

that f1|a2 Dn-1 is homotopic to f2|a2 Dn-1 in a2xY-K modulo a2 Sn-2. f1, f2 satisfy

all the conditions of Lemma 2. Hence there is an

R-preserving

map

Fi

of

Al u A2

into R x Y-K such that

F1|03C3

= g and

Fi =/1

on

6(al ).

Let

f’0 = f0, f’1 = F1|A1 and f’2 = F11A2. Thus f’0|03C3(a1) = f’1|03C3(a1) and f’1|03C3(a2) = f’2|03C3(a2). Repeat

the above process

for/2 and 13 .

It follows that there are

R-preserving maps f"2 and f’3

defined

respectively

on

A2

and

A3

such that

both agree

with g

on

03C3, f"2|03C3(a2) = f’1|03C3(a2), f"2|03C3(a3) = f’3|03C3(a3)

and so on.

Define

Since

{Ai}

is

finite,

it is clear that we can extend G to the entire

03C3 = A0 ~ A1 ~ ··· ~ Am

with the

required properties.

CASE 2. n = 1. For each

(t, y)

E R x

Y-K,

there is an open interval

L, containing t

such that

L,

x y ~ K =

0.

Since the inclusion t x Y-K ~ t x Y is a

homotopy equivalence,

K cannot contain t x Y. It follows that

the set of all

Lt

covers

[a, b]. By compactness,

there are a finite number

a = a0 a1 ··· am+1 = b and a finite subset

{u0, u1, ···, um}

of

Y such that for

all i, [ai, ai+1] ui nK= 0, (ao, uo)

=

g(03C3(a))

and

(am+1, um)

=

9 ( u (b ) ).

Let

Ai

=

U {03C3(t)|ai ~ t ~ ai+1}, i

=

0, ···, m

and

define fi : Ai ~

R x Y-K

by fi(03C3(t)) = (t, u;).

We may assume

fola(a1)

=

f1|03C3(a1) and fm-1|03C3(am)

=

fm|03C3(am).

The rest of the

proof

is the

same as Case 1.

We are now

ready

to state our main Lemma.

LEMMA 4. Let E be a metric TVS and Y a metric AR. Let K be a closed subset

of

R x Y such that each K n t x Y is

weakly homotopy negligible

in

t x Y. Let

ITI

be a

finite simplicial complex

in R x E and S a

subcomplex of T.

Then each

R-preserving

map

of |S|

into R x Y-K can be extended

to an

R-preserving

map

of 1 Tl

into R x Y- K.

Furthermore,

the same is

true when ’R’ is

replaced by

’an interval

of

R’.

PROOF. Denote the i-skeleton of T

by Ti (that is,

all

simplices

of T of

dimension ~ i). Let f o

be an

R-preserving of 19 into

R x Y- .K. Since

t x Y is not contained in K for any t, we may

assume fo

is an

R-preserving

map

of |S| ~ |T0|

into R Y-K.

By

Lemma

3, fo

can be extended to

an

R-preserving map fl of |S| u |T1| into

R x Y-K.

By

Lemma 3

again

we can

extend fl

to an

R-preserving map f2 of 19 u T2

into R x Y- K.

Inductively, fo

extends to an

R-preserving

map

of 1 Tl

into R x Y-K.

(9)

4. Proofs

PROOF oF THEOREM 1. We need

only

to show

(see Proposition 1)

that

K is

locally homotopy negligible

in R’ x X. Let U b; a closed

neighbor-

hood of x E X such that there is a

homeomorphism ~

of U onto a closed

neighborhood

V of C. Let J be a closed interval of R’. It follows that J x U is contractible and we want to show

03C0k(J U-K)

= 0 for all

k ~

1. So

suppose f : (Dn, Sn-1) ~ (J U, J U-K)

is a map.

Let 03C8

be the

identity

map of J onto J and let =

(03C8, ~) :

J x U - J x V. Let

f0 = 03BB·f

and

Ko = 03BB(K ~ J U). Ko

is closed in R V and

f0 :

(Dn, Sn-1) ~ (J V, J V-K0).

J V is a convex subset of R x E.

By

standard

argument

it follows

that fo

is

homotopic

to a

simplical

map g of

(Dn, Sn-1)

into

(J x V, J V-K0) by maps ffl,

t such that each

ft : Dn ~ J V, f1 = g and ft(Sn-1)

n

K0 = 0

for all t.

By hypothesis

of K and

Proposition 1,

it follows that

K0

n t x V is

homotopy negligible

in t V for each x E R. Since V is an

AR, by

Lemma 4 there is a map y of

g(Dn)

into Jx

V-Ko

such that

03BC|g(Sn-1).

=

identity.

Let h = ,ug.

Combining

with the

homotopy {ft}

we therefore have shown that there is a map

Fo :

nn -+ J x

h- Ko

such that

Fo Is. -, = fo.

Let F =

03BB-1F0.

Thus F : D n --+ J x U- K and

F|Sn-1 = f.

This shows J x U- K is

weakly homotopy negligible. By locally convexity of E, {J U}

form a

neighbor-

hood

system

of R’ x X. The theorem now follows from

Proposition

1.

PROOF oF THEOREM 2. Let

Ul, U2

be closed

neighborhood

of x1 ~

X1,

x2 E

X2 respectively

such that for i =

1, 2,

there are

homeomorphisms

~i of

Ui

onto a convex closed

neighborhood Vi

of

Ci,

where

Ci

is a closed

convex subset of a TVS

Ei. By

local

convexity

of

El

and

E2, {U1 U2}

form a

neighborhood

system of

Xl x X2,

so

by Proposition 1,

we need

only

to show

1Ck(Ul

x

U2-K)

= 0 for

all k ~

1.

Let f : (Dn, Sn-1)

~

( Ul

x

U2 , Ul

x

U2-K) be

a map and

let f0 = 03BBf,

where 2 =

(~1, ~2). So fo : (Dn, Sn-1) ~ (V1 V2, V1 V2-K0)

where

K0 = 03BB(K ~ U1

x

U2).

It suffices to show that there is a map

Fo :

Dn ~

Vl

x

V2-K0

such that

F0|Sn-1 = f0.

As in Theorem

1,

it is well-known that there is a

homotopy {ft}

of map of Dn into

V, x V2

such that

ft(Sn-1)

~

Ko = 0

for all t and g =

fl is simplicial.

Let F be the finite-

dimensional

subspace

of

El

x

E2 generated by g(Dn).

Let

F, = Pi (F)

where

P1 : El

x

E2 -+ El

is the

projection. Ko

is closed in

El

x

V2, by hypothesis

and

Proposition 1, x x V2

n

Ko

is

locally homotopy negligible

in

x x Tl2

for each

x E El . By

Theorem 1

K0 ~ (Jl x Tl2)

is

strongly homotopy negligible

in

J1 V2

for any

1-simplex J1

of

El.

Since

Fl

is

finite-dimensional,

say of

dimension n, by induction, K0

n

(Jm

x

Tl2)

is

strongly homotopy negligible

in

Jl

x

V2

for any m-cube Jm in

Fl .

K1 = P1g(Dn)

is an

m-simplex

of

Fi,

hence

homeomorphism

to some

(10)

127

m-cube Jm. It follows that

K0

n

(Ki

x

Tl2 )

is

strongly homotopy negli- gible

in

K1

x

V2,

which contains

g(Sn-1).

Thus

g|Sn-1

can be extended

to a gl of D" into

(K1

x

v2) - Ko . Combining

with the

homotopy {ft},

we have shown that there is a map

F0

of Dn into

V1 V2-K

such

that, F0|Sn-1

=

f0.

The

proof

of the theorem is now

complete.

PROOF oF THEOREM 4. Let U be a canonical closed

neighborhood

of I"

that

is,

U is a

product

of closed intervals. Let

U1 =

U- sn. Let V be a closed

neighborhood

of X such that there is a

homeomorphism

of V

onto a closed convex subset

Vl

of a TVS E. For

simplicity

we assume

V1

is V. It follows from the

hypothesis

and from Theorem 3 that

K ~

( Ul

x

V)

is

homotopy negligible in Ul

x V. Now

let f : (Dn, Sn-1)

~

(U x V,

U x

V-K)

be a map. Since

f(Sn-1) ~ K = Ø,

let 03B5 &#x3E; 0 denote the distance

between f(Sn-1)

and K

(distance

in the sense of the

usual induced metric on the

product

space I n x

X).

It is evident that there is a

homotopy {gt}

of maps of Un Sn into U such that go =

identity, gl(U n sn)

c

Ul

and

d(gt, go)

8 for all t. Now define a

homotopy

{ft}

of maps of U x V into itself

by ft(x, y) = (gt(x), y).

It follows that

fo = identity, fl ( U x V)

c

Ul

x V

and ft(Sn-1) ~ K = Ø

for all t. Let

ht = ft f.

Then

h 1 : (Dn, Sn-1) ~ ( Ul V, U1 V-K).

Thus

hl fsn-l

ex- tends to a map h of Dn into

Ul

x V- K.

Combining

with the

homotopy {ht},

we have shown

that f|Sn-1

extends to a map of Dn into U x V-K.

Thus U x V-K is

homotopically

trivial and the

proof

of the theorem is

complete.

PROOF oF THEOREM 5.

By

the

Corollary

of

[3-Thm. 15] it

suflices to

show that

Ci

x

C2-K is homotopically

trivial.

We first consider the

special

case that

Cl

is a closed convex subset of

the reals R. Let

f : (Dn, Sn-1) ~ (Cl

x

C2, Cl

x

C2 - K)

be a map. As illustrated in the

proofs

of the

previous

theorems

(which

are rather

elementary)

we may assume

f

is

simplicial. By

Lemma 4 there is a

C1-preserving

map g

of f(Dn)

into

el x C2 - K

such that

g 1 f(s. - i)

=

identity.

Hence

Cl x C2 -K

is

homotopically

trivial and the

proof

of

the

special

case is

complete.

Now consider the

general

case. The

proof

is very similar to that of Theorem 2. We

proceed

as follows.

Let f : (Dn, Sn-1) ~ (C1 C2,

C1

x

C2-K)

be a map.

Again

we may

assume f is simplicial.

Let F be

the finite dimensional

subspace

of

El

x

E2 generated by f(Dn).

Let

F1 = P1(F)

where

P1 : E1 E2 ~ E1

is the

projection. By

the

special

case proven

above, K

n

(Jl

x

C2)

is

homotopy negligible

in

Jl

x

C2

for

each

1-simplex Jl of El. Inductively,

K ~

(Jn C2)

is

homotopy negli- gible

in any n-cube of

Fi.

Since

K1 = P1(f(Dn))

is an

m-simplex,

which

is

homeomorphic

to an

m-cube,

it follows that n

(Ki

x

C2)

is homo-

(11)

topy negligible

in

K1 C2,

which

contains f(Sn-1).

Hence we may make

the extension

of f|Sn-1

to Dn in

K1

x

C2-K

and the

proof

is

complete.

PROOF oF THEOREM 6. The

proof

makes use of Theorem 5 and is

exactly

the same as that of Theorem 4.

REFERENCES

R. D. ANDERSON

[1] On topological infinite deficiency, Mich. Math. J. 14 (1967), 365-383.

J. EELLS, Jr. and N. H. KUIPER

[2] Homotopy negligible subsets, Compositio Math. 21 (1969) 155-161.

[3] R. S. PALAIS

Homotopy theory of infinite dimensional manifolds, Topology 5 (1966) 1-16.

R. D. ANDERSON, D. HENDERSON, and J. WEST

[4] Negligible subsets of infinite-dimensional manifolds, Compositio Math. 21 (1969)

143-150.

[5] Problems in Infinite-dimensional spaces and manifolds, Manuscript, LSU, Baton Rouge, La. 1969.

[6] On homeomorphisms of infinite-dimensional bundles, I, to appear.

(Oblatum 18-111-71) Department of Mathematics University of California Santa Barbara, Calif. 93106 U.S.A.

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